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Mirrors > Home > ILE Home > Th. List > Mathboxes > redcwlpo | Unicode version |
Description: Decidability of real
number equality implies the Weak Limited Principle
of Omniscience (WLPO). We expect that we'd need some form of countable
choice to prove the converse.
Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 13774). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones. Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO". WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10172 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.) |
Ref | Expression |
---|---|
redcwlpo | DECID WOmni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . . 6 DECID DECID | |
2 | elmapi 6627 | . . . . . . . . 9 | |
3 | 2 | adantl 275 | . . . . . . . 8 DECID |
4 | oveq2 5844 | . . . . . . . . . . 11 | |
5 | 4 | oveq2d 5852 | . . . . . . . . . 10 |
6 | fveq2 5480 | . . . . . . . . . 10 | |
7 | 5, 6 | oveq12d 5854 | . . . . . . . . 9 |
8 | 7 | cbvsumv 11288 | . . . . . . . 8 |
9 | 3, 8 | trilpolemcl 13757 | . . . . . . 7 DECID |
10 | 1red 7905 | . . . . . . 7 DECID | |
11 | eqeq1 2171 | . . . . . . . . 9 | |
12 | 11 | dcbid 828 | . . . . . . . 8 DECID DECID |
13 | eqeq2 2174 | . . . . . . . . 9 | |
14 | 13 | dcbid 828 | . . . . . . . 8 DECID DECID |
15 | 12, 14 | rspc2v 2838 | . . . . . . 7 DECID DECID |
16 | 9, 10, 15 | syl2anc 409 | . . . . . 6 DECID DECID DECID |
17 | 1, 16 | mpd 13 | . . . . 5 DECID DECID |
18 | 3, 8 | redcwlpolemeq1 13774 | . . . . . 6 DECID |
19 | 18 | dcbid 828 | . . . . 5 DECID DECID DECID |
20 | 17, 19 | mpbid 146 | . . . 4 DECID DECID |
21 | 20 | ralrimiva 2537 | . . 3 DECID DECID |
22 | nnex 8854 | . . . 4 | |
23 | iswomninn 13770 | . . . 4 WOmni DECID | |
24 | 22, 23 | ax-mp 5 | . . 3 WOmni DECID |
25 | 21, 24 | sylibr 133 | . 2 DECID WOmni |
26 | nnenom 10359 | . . 3 | |
27 | enwomni 7125 | . . 3 WOmni WOmni | |
28 | 26, 27 | ax-mp 5 | . 2 WOmni WOmni |
29 | 25, 28 | sylib 121 | 1 DECID WOmni |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 DECID wdc 824 wceq 1342 wcel 2135 wral 2442 cvv 2721 cpr 3571 class class class wbr 3976 com 4561 wf 5178 cfv 5182 (class class class)co 5836 cmap 6605 cen 6695 WOmnicwomni 7118 cr 7743 cc0 7744 c1 7745 cmul 7749 cdiv 8559 cn 8848 c2 8899 cexp 10444 csu 11280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-2o 6376 df-oadd 6379 df-er 6492 df-map 6607 df-en 6698 df-dom 6699 df-fin 6700 df-womni 7119 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-ico 9821 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-ihash 10678 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 |
This theorem is referenced by: (None) |
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